What's the substance of the controversy? The Abrams et al. group frames the issue as an opportunity to reduce the harms of cigarette smoking. They write:

"Inhalation of the toxic smoke produced by combusting tobacco products, primarily cigarettes, is the overwhelming cause of tobacco-related disease and death in the United States and globally. A diverse class of alternative nicotine delivery systems (ANDS) has recently been developed that do not combust tobacco and are substantially less harmful than cigarettes. ANDS have the potential to disrupt the 120-year dominance of the cigarette and challenge the field on how the tobacco pandemic could be reversed if nicotine is decoupled from lethal inhaled smoke. ANDS may provide a means to compete with, and even replace, combusted cigarette use, saving more lives more rapidly than previously possible. ... A reframing of societal nicotine use through the lens of harm minimization is an extraordinary opportunity to enhance the impact of tobacco control efforts."

Here's a figure from their paper showing how they perceive the harms from e-cigarettes: that is, similar to other aids to quitting smoking like nicotine patches or nasal sprays, clearly lower than smokeless tobacco, and dramatically lower than combusted tobacco.

As Glantz and Bareham see it, most e-cig user are already smokers. The health gains from e-cigs over conventional smoking are probably real, but not all that large. So if e-cigs lead to increasing use of tobacco products (that is, conventional smoking and vaping combined), the overall result could be negative for public health.

Here's a figure from their paper showing patterns of teens and e-cigarettes. Notice that conventional smoking has dropped faster than expected since the arrival of vaping, but the overall downward trend--combining smoking and vaping--maybe altering.

Callier sums up these patterns in the overview essay for Knowable:

In a meta-analysis of 20 studies, Glantz and his UCSF colleague Sara Kalkhoran found that the odds of quitting cigarettes were 28 percent lower for smokers who used e-cigarettes than for those who did not. In the real world, many e-cigarette users take them up with an intention to quit tobacco. But others use them without such an aim, perhaps to get a nicotine fix in areas with smoking restrictions. “Importantly, most adults who use e-cigarettes continue to smoke conventional cigarettes (referred to as dual users),” Glantz and David Bareham of Lincolnshire Community Health Services in the UK wrote in the 2018 Annual Review of Public Health. “In 2014 in the United States, 93 percent of e-cigarette users continued to smoke cigarettes, 83 percent in France, and 60 percent in the United Kingdom.”

In terms of health issued, Glantz and Kalkhoran cite studies that raise concerns over whether vaping reduced the health consequences for hearts and lungs: in particular, they raise questions over whether ultrafine particles created by vaping may pose health risks.

"Survey data from the charity ASH (Action on Smoking and Health) indicates that the vast majority of e-cigarette users in the UK are either ex-smokers or current smokers,5 and regular use among ‘never smokers’ remains very low, at less than one per cent.3,5 The level of e-cigarette use by children also remains low ..."

On the health issue, the BMA writes that also the evidence on vaping is still accumulating (especially evidence on long-term effects), the weight of the evidence at present is that nicotine alone is substantially better than nicotine-and-smoke. Moreover, vaping has become the most popular method of trying to quit smoking. The report argues (footnotes omitted):

"For example, though long-term inhalation of nicotine vapour is associated with some level of risk, and there continues to be debate over the precise level of this risk, several reviews have concluded that it is substantially lower than inhaling tobacco smoke.While NICE guidance states that smokers should always be advised that stopping smoking in one step is the best approach, it also recommends advising those who do not want, are not ready or are unable to stop smoking in one step, to consider a harm-reduction approach.It is the tar and other toxins in tobacco smoke, rather than nicotine, that are responsible for most of the harm associated with smoking. ...

"In a 2016 consensus statement coordinated by PHE (Public Health England) a range of health organisations together stated that “the evidence suggests that the health risks posed by e-cigarettes are relatively small by comparison” [to smoking]. Similarly, a 2017 consensus statement from NHS Health Scotland – endorsed by a range of health organisations – stated that e-cigarettes are “definitely less harmful than smoking tobacco”.Although there is consensus that e-cigarettes are less harmful than smoking, and that any risks associated with their use are likely to be significantly lower than tobacco, quantifying the precise level of this risk is complex. The most widely cited estimate of relative risk is from PHE’s 2015 e-cigarette evidence review – which concluded that it would be reasonable to estimate that e-cigarette use is likely to be around 95% safer than smoking. This figure was endorsed by the RCP (Royal College of Physicians) in their 2016 report Nicotine without smoke: Tobacco harm reduction, which concluded that “…the hazard to health arising from long-term vapour inhalation is unlikely to exceed 5% of the harm from smoking tobacco smoke” ...

"Significant numbers of smokers are now using e-cigarettes in attempts to stop smoking. The most recent data from the smoking toolkit study indicates that 34% of people trying to stop smoking use an e-cigarette, and e-cigarettes are the most popular device used in attempts to quit smoking. ... Overall – while there is a lack of high-quality research into their effectiveness as a cessation aid – most reported studies demonstrate a positive relationship between e-cigarette use and smoking cessation."

It's hard for an outsider like me, without training in medicine and epidemiology, to evaluate these controversies. On one side, there is a danger that vaping could lead to an increase in tobacco use, and to health harms that are not yet understood. On the other hand, there is a danger that sharp restrictions on vaping would lead more people to continue with conventional cigarettes, with health harms that are very well understood.

For what it's worth, my reading of the evidence comes down somewhat in favor of a belief that vaping will disrupt the cigarette industry, and that the probabilities are in favor of believing that this change will benefit public health. To put it another way, efforts to treat vaping as if it is fully equivalent to conventional cigarettes run a real risk of costing lives.